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A227310 G.f.: 1/G(0) where G(k) = 1 + (-q)^(k+1) / (1 - (-q)^(k+1)/G(k+1) ). 11
1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 1, 3, 2, 3, 4, 4, 6, 7, 8, 11, 13, 16, 20, 24, 31, 37, 46, 58, 70, 88, 108, 133, 167, 204, 252, 315, 386, 479, 594, 731, 909, 1122, 1386, 1720, 2124, 2628, 3254, 4022, 4980, 6160, 7618, 9432, 11665, 14433, 17860, 22093, 27341, 33824, 41847, 51785, 64065, 79267 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,13

COMMENTS

Number of rough sand piles: 1-dimensional sand piles (see A186085) with n grains without flat steps (no two successive parts of the corresponding composition equal), see example. - Joerg Arndt, Mar 08 2014

The sequence of such sand piles by base length starts (n>=0) 1, 1, 0, 1, 0, 2, 0, 5, 0, 14, 0, 42, 0, ... (A097331, essentially A000108 with interlaced zeros).  This is a consequence of the obvious connection to Dyck paths, see example. - Joerg Arndt, Mar 09 2014

a(n>=1) are the Dyck paths with area n between the x-axis and the path which return to the x-axis only once (at their end), whereas A143951 includes paths with intercalated touches of the x-axis. - R. J. Mathar, Aug 22 2018

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Joerg Arndt)

A. M. Odlyzko and H. S. Wilf, The editor's corner: n coins in a fountain, Amer. Math. Monthly, 95 (1988), 840-843.

FORMULA

a(0) = 1 and a(n) = abs(A049346(n)) for n>=1.

G.f.: 1/ (1-q/(1+q/ (1+q^2/(1-q^2/ (1-q^3/(1+q^3/ (1+q^4/(1-q^4/ (1-q^5/(1+q^5/ (1+-...) )) )) )) )) )).

G.f.: 1 + q * F(q,q) where F(q,y) = 1/(1 - y * q^2 * F(q, q^2*y) ); cf. A005169 and p. 841 of the Odlyzko/Wilf reference; 1/(1 - q * F(q,q)) is the g.f. of A143951. - Joerg Arndt, Mar 09 2014

G.f.: 1 + q/(1 - q^3/(1 - q^5/(1 - q^7/ (...)))) (from formulas above). - Joerg Arndt, Mar 09 2014

G.f.: F(x, x^2) where F(x,y) is the g.f. of A239927. - Joerg Arndt, Mar 29 2014

a(n) ~ c * d^n, where d = 1.23729141259673487395949649334678514763130846902468... and c = 0.0773368373684184197215007198148835507944051447907... - Vaclav Kotesovec, Sep 05 2017

G.f.: A(x) = 2 -1/A143951(x). - R. J. Mathar, Aug 23 2018

EXAMPLE

From Joerg Arndt, Mar 08 2014: (Start)

The a(21) = 7 rough sand piles are:

:

:   1:      [ 1 2 1 2 1 2 1 2 1 2 3 2 1 ]  (composition)

:

:           o

:  o o o o ooo

: ooooooooooooo  (rendering of sand pile)

:

:

:   2:      [ 1 2 1 2 1 2 1 2 3 2 1 2 1 ]

:

:         o

:  o o o ooo o

: ooooooooooooo

:

:

:   3:      [ 1 2 1 2 1 2 3 2 1 2 1 2 1 ]

:

:       o

:  o o ooo o o

: ooooooooooooo

:

:

:   4:      [ 1 2 1 2 3 2 1 2 1 2 1 2 1 ]

:

:     o

:  o ooo o o o

: ooooooooooooo

:

:

:   5:      [ 1 2 3 2 1 2 1 2 1 2 1 2 1 ]

:

:   o

:  ooo o o o o

: ooooooooooooo

:

:

:   6:      [ 1 2 3 2 3 4 3 2 1 ]

:

:      o

:   o ooo

:  ooooooo

: ooooooooo

:

:

:   7:      [ 1 2 3 4 3 2 3 2 1 ]

:

:    o

:   ooo o

:  ooooooo

: ooooooooo

(End)

From Joerg Arndt, Mar 09 2014: (Start)

The A097331(9) = 14 such sand piles with base length 9 are:

01:  [ 1 2 1 2 1 2 1 2 1 ]

02:  [ 1 2 1 2 1 2 3 2 1 ]

03:  [ 1 2 1 2 3 2 3 2 1 ]

04:  [ 1 2 1 2 3 2 1 2 1 ]

05:  [ 1 2 1 2 3 4 3 2 1 ]

06:  [ 1 2 3 2 1 2 3 2 1 ]

07:  [ 1 2 3 2 1 2 1 2 1 ]

08:  [ 1 2 3 2 3 2 1 2 1 ]

09:  [ 1 2 3 2 3 2 3 2 1 ]

10:  [ 1 2 3 4 3 2 1 2 1 ]

11:  [ 1 2 3 2 3 4 3 2 1 ]

12:  [ 1 2 3 4 3 2 3 2 1 ]

13:  [ 1 2 3 4 3 4 3 2 1 ]

14:  [ 1 2 3 4 5 4 3 2 1 ]

(End)

PROG

(PARI) N = 66;  q = 'q + O('q^N);

G(k) = if(k>N, 1, 1 + (-q)^(k+1) / (1 - (-q)^(k+1) / G(k+1) ) );

gf = 1 / G(0);

Vec(gf)

(PARI)

N = 66;  q = 'q + O('q^N);

F(q, y, k) = if(k>N, 1, 1/(1 - y*q^2 * F(q, q^2*y, k+1) ) );

Vec( 1 + q * F(q, q, 0) ) \\ Joerg Arndt, Mar 09 2014

CROSSREFS

Cf. A049346 (g.f.: 1 - 1/G(0), where G(k)= 1 + q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).

Cf. A226728 (g.f.: 1/G(0), where G(k) = 1 + q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).

Cf. A226729 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+2) ) ).

Cf. A006958 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+1)/G(k+1) ) ).

Cf. A227309 (g.f.: 1/G(0), where G(k) = 1 - q^(k+1) / (1 - q^(k+2)/G(k+1) ) ).

Cf. A173258, A291874.

Sequence in context: A059779 A291874 A049346 * A291905 A240853 A319849

Adjacent sequences:  A227307 A227308 A227309 * A227311 A227312 A227313

KEYWORD

nonn

AUTHOR

Joerg Arndt, Jul 06 2013

STATUS

approved

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Last modified May 26 01:22 EDT 2020. Contains 334613 sequences. (Running on oeis4.)